Decomposition of loop spaces and periodic problem on $\pi_*$

TL;DR

This paper studies the structure of loop spaces and periodic problems in homotopy groups, providing a localized family of spaces at 2 with specific stable and unstable homotopy group relationships, including applications to mod 2 Moore spaces.

Contribution

It introduces a new family of localized spaces at 2 where stable homotopy groups are summands of unstable ones, advancing understanding of homotopy decompositions.

Findings

Stable homotopy groups are summands of unstable groups for the constructed spaces.

Application to mod 2 Moore spaces demonstrates practical relevance.

Provides a framework for analyzing periodic problems in homotopy theory.

Abstract

We provide a family of spaces localized at 2, whose stable homotopy groups are summands of their unstable homotopy groups. Application to mod 2 Moore spaces are given.